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REVIEW ON TODD EISWORTH’S CHAPTER
FOR THE HANDBOOK OF SET THEORY:
“SUCCESSORS OF SINGULAR CARDINALS”
ASSAF RINOT
This chapter offers a comprehensive and lucid exposition of the questions and techniques involved
in the study of combinatorics of successors of singular cardinals.
What is so special about successors of singular cardinals? They are successor cardinals, but are
also similar to inaccessible cardinals, in the sense that there is no maximal regular cardinal below
them, meaning for instance that there is no obvious obstruction for each of their stationary sets
to reflect.1 And indeed, Magidor proved [Mag82] that, modulo a large cardinal hypothesis, it is
consistent that every stationary subset of ℵω+1 reflects (the proof may be found as Theoreom 2.15
of this chapter).
But even in the absence of large cardinals, singular cardinals and their successors are tightly
related. Recall that by König’s theorem [Kön05], the collection of all countable subsets of the
(highly uncountable) cardinal ℵω has cardinality at least the successor of ℵω . In fact, all reasonable
finer measures (such as density [Koj15], and cofinality with respect to inclusion) of the collection
[ℵω ]ω of all countable subsets of ℵω measure it by at least ℵω+1 . Arguably, a canonical witness
to König’s theorem is a pcf scale. A pcf scale is a pair (~
µ, f~), where µ
~ = hµi : i < σi is an
increasing sequence of regular cardinals, converging to some singular cardinal µ of cofinality σ, and
f~ = hfα : α < µ+ i is a sequence of functions such that:
Q
(1) for all α < µ+ , fα ∈ i<σ µi ;
(2) for all α < Q
β < µ+ , D(fα , fβ ) := {i < σ | fα (i) ≥ fβ (i)} is bounded in σ;
(3) for all f ∈ i<σ µi , there exists some β < µ+ , such that D(f, fβ ) is bounded in σ.
By a theorem of Shelah [She94, p. 50], every singular cardinal µ admits such a pcf scale, and the
proof may be found as Theorem 3.53 of this chapter.
Next, we briefly mention a few differences between successors of singular cardinals and successors
of regular cardinals:
• By a theorem of Specker [Spe49], GCH entails that for every regular cardinal µ, there exists
a µ+ -Aronszajn tree, that is, a tree of height µ+ without chains or levels of size µ+ . On the
other hand, Magidor and Shelah [MS96] proved that, modulo a large cardinal hypothesis,
GCH is consistent with the nonexistence of an ℵω+1 -Aronszajn tree (cf. [Sin12], [Nee14]).
µ+
+
• By a theorem of Shelah [She91], for every regular cardinal µ, the set E6=
cf(µ) := {α < µ |
cf(α) 6= cf(µ)} is in the approachability ideal I[µ+ ] (Corollary 4.6 of this chapter), whereas
by another theorem of Shelah [She79], modulo a large cardinal hypothesis, it is consistent
µ+
+
that for some singular cardinal µ, E6=
cf(µ) is not in I[µ ] (Theorem 3.20 of this chapter).
The definition of the approachability ideal I[κ] may be found in Definition 3.3 of this
chapter, but we also include it in here: A subset S ⊆ κ is in I[κ] iff there exists a club
C ⊆ κ and a sequence hai | i < κi satisfying the following. For every α ∈ S ∩ C, there exists
a cofinal subset A ⊆ α of order-type cf(α) < α such that {A ∩ β | β < α} ⊆ {aβ | β < α}.
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Recall that a stationary set S is said to reflect if there exists some ordinal γ < sup(S) of uncountable cofinality
for which S ∩ γ is stationary in γ.
1
• Recall that an algebra A is said to be a Jónsson algebra if all of its proper subalgebras have
smaller cardinality than that of A. For instance, the construction of a Jónsson group of
cardinality ℵ1 may be found in [She80].
By a theorem of Tryba [Try84] and independently, Woodin, if κ is the successor of a
regular cardinal, then there exists a Jónsson algebra of cardinality κ (Corollary 5.5 of this
chapter). Whether the same statement holds true for successors of singular cardinals is a
long-standing open problem (see [Eis12] for the best known partial answer).
Below, we shall review the content of each section, and mention a few relevant papers that
appeared after the chapter was written.
I Section 1 provides some necessary background and motivation.
I Section 2 deals with some of the possible behaviors of stationary reflection. Specifically,
Jensen’s square principle µ (Definition 2.1 of this chapter) is a strong anti-reflection principle
while, on the other extreme, large cardinals (such as supercompact cardinals, see Definition 1.20)
give rise to reflection, and even to combinatorial objects (such as indecomposable ultrafilters, see
Definition 2.7) that entail reflection.
As mentioned earlier, in [Mag82] Magidor established the consistency of the statement “every
stationary subset of ℵω+1 reflects”. Magidor’s original proof included a component of iterated clubshooting that was later eliminated by an idea of Shelah that had to do with the approachability
ideal I[ℵω+1 ]. The proof given in [Mag82], as well as the one in this chapter, is Shelah’s and
serves as a first exposition to the approachability ideal. It is a curious fact that many years later,
Magidor’s original approach was found useful in getting finer models of reflection (see [CFM01]). A
recent result concerning a fine form of reflection at successors of singulars may be found in [CLH16].
I Section 3 offers a thorough study of the approachability ideal I[λ], and its weaker sibling I[λ; µ].
Various equivalent characterizations of these ideals (stemming from [She79], [She93], [FM97]) are
presented. The negative effects of large cardinals on the approachability ideal are discussed (Theorem 3.20). In the positive direction, a proof is given of Shelah’s key theorem that for regular
cardinals κ, σ, λ satisfying κ+ < σ < λ, the ideal I[λ] contains a stationary subset of Eκλ that
reflects (also) at points of cofinality σ (Theorem 3.18). The proof relies on Shelah’s club guessing
principle, a concept that makes a second appearance in Section 5 of this chapter. Under the hypothesis that every stationary subset of ℵω+1 reflects, it is proved that if ℵω+1 is a strong limit,
then ℵω+1 ∈ I[ℵω+1 ] (Corollary 3.41). See [SV10] for a study of the effect of reflection principles
at small regular cardinals on the approachability ideal at the successor of singular cardinals.
The second part of Section 3 deals with pcf scales, and the interrelation of sets in the approachability ideal with the sets of good points in such scales (see Theorem 3.55 of this chapter, as well
as [CFM04], [GS08], [CF10]). A motivation for studying good points may be found, for instance,
in [MS94] and [Cum97].
I Section 4 is dedicated to variations of Jensen’s square principle µ , their effect on reflection
of stationary sets, Jensen’s diamond principle (Definition 4.22 of this chapter), and pcf scales. For
more recent results, see [AKY09], [CM11], [Sak13], [TTP14], [Sak15].
The section offers a clear proof of an old and difficult theorem of Shelah from [She84] (Theorem 4.24 of this chapter). Incidentally, shortly before this Handbook went to print, Shelah found a
simple (yet, ingenious) proof of a strong theorem concerning diamond [She10], and this was quickly
adapted to the case of successors of singulars by Zeman [Zem10] (See also [Rin10], [Mat14], [Rin15]).
The second part of Section 4 deals with special types of pcf scales: good, very good, better and
their applications to incompactness combinatorics such as the NPT and ADS principles (Definitions
4.44 and 4.71 of this chapter, respectively), simultaneous reflection of stationary sets, and reflection
of generalized stationary sets (See also [Cum05], [She08], [FJS+ 10]).
I Section 5 is a short section dedicated to results around the question of whether the successor
of a singular cardinal must carry a Jónsson algebra.
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Two theorems that allow one to infer the existence of a Jónsson algebra on some cardinal from the
existence of Jónsson algebras on cardinals below it, are given. The second among these theorems
motivates the study of Shelah’s club guessing principle (see also [ES09], [Eis10], [Rin14c]). A related
concept is the negative partition relation µ+ 9 [µ+ ]2θ that asserts the existence of a coloring of
unordered pairs c : [µ+ ]2 → θ with the property that for every cofinal A ⊆ µ+ and every i < θ,
there exists some α < β from A for which c(α, β) = i. A further generalization is the principle
Pr1 (µ+ , θ, σ) that coincides with the former in the case σ = 2. The earliest demonstrations of the
utility of the case σ > 2 appeared in [Roi78] and [Gal80].
Shelah proved that Pr1 (µ+ , cf(µ), cf(µ)) holds for every singular cardinal µ, and this may be
found as Theorem 5.16 of this chapter. Extending a theorem of Moore [Moo06] (which extended a
theorem of Todorcevic [Tod87]), Peng and Wu recently proved that Pr1 (ℵ1 , ℵ1 , n) holds for every
positive integer n. For additional recently-published results in this direction, see [Eis12], [Rin12],
[Eis13a], [Eis13b], [Rin14a], [Rin14b].
Upon the advice of the editor, we also include a short list of well-known open problems:
(1) Is it consistent that µ+ 9 [µ+ ]2µ fails for some singular cardinal µ?
ℵ
ℵ
(2) It is consistent that Eℵ2ω+1 6∈ I[ℵω+1 ] or Eℵ3ω+1 6∈ I[ℵω+1 ]? What about the consistency of
(ℵω+1 , ℵω ) (ℵ2 , ℵ1 ) or (ℵω+1 , ℵω ) (ℵ3 , ℵ2 ) ?
µ+
(3) Is it consistent that GCH holds and ♦(Ecf(µ)
) fails for some singular cardinal µ?
(4) Is it consistent that for some singular cardinal µ, and a reflecting stationary subset S ⊆ µ+ ,
NSµ+ S is saturated?
(5) Suppose that 2ℵn < ℵω for all n < ω, and 2ℵω > ℵω+1 .
Must there exist an ℵω+1 -Aronszajn tree?
(6) Suppose that µ is a singular cardinal and there exists a special µ+ -Aronszajn tree.
Does GCH entail the existence of a µ+ -Souslin tree?
(7) Suppose that µ is a singular cardinal of countable cofinality.
Must there exist a matrix hξα,n | α < µ+ , n < ωi such that for every club C in µ+ , there
exists some α < µ+ such that the following two conditions hold simultaneously?
(a) hξα,n | n < ωi is a strictly increasing sequence of ordinals from C, converging to α;
(b) hcf(ξα,n ) | n < ωi is a strictly increasing sequence of cardinals, converging to µ.
In summary, this chapter is an invaluable introduction and invitation to the subject.
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Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel.
URL: http://www.assafrinot.com
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